theoretical and experimental results on the interaction of two vortex rings are .... the vortex rings undergo a head-on collision and the ow maintains its symmetry.
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Application of 3D Particle Method to the Prediction of Aerodynamic Sound K.P. Pothou, S.G. Voutsinas NTUA, P.O. Box 64070 15710, Zografou Athens, Greece
S.G. Huberson GEMH, Universite du Havre, B.P. 540 76058 Le Havre cedex France
O.M. Knio Dept. of Mechanical Engineering The Johns Hopkins University Baltimore MD 21218, USA
Abstract
Low-Mach-number aeroacoustic theories are applied to predict far- eld sound emission from vortex-dominated ows. The latter are simulated using three-dimensional vortex methods. Both particle schemes and lament-based schemes are used. Comparisons of model predictions with theoretical and experimental results on the interaction of two vortex rings are rst discussed. Next, the schemes are used to examine the e ect of initial conditions and 3D perturbations on the properties of the far- eld acoustic pressure. Finally, a more elaborate wind-turbine model is presented and its application to the evaluation of several turbine-blade designs is summarized.
1 Governing equations and vortex sound theory Noise is a small pressure perturbation related to the compressibility of uids and its modelling must be based on the compressible Navier-Stokes equations. Although the full Navier-Stokes equations can be numerically solved, the complexity of the problem makes them very dicult to use in general. The main problems encountered are the derivation of noise-compatible external boundary conditions, and the usual need of very ne grids. In noise simulations, the problem is even more crucial since the grid is directly related to the lowest resolvable scale, and thus to the highest frequency which is accurately represented. An obvious way to overcome these diculties is to introduce more physics in the problem in order to obtain a simpli ed formulation. This can be achieved by using a few more assumptions which are satis ed in most cases of audible noise. We rst assume that the noise sources are contained in a nite domain. In our case, the noise sources are mainly vorticity-generated pressure waves; thus, this assumption is equivalent to the requirement that the support of vorticity is compact. As a result, the ow eld can be split into an inner and an outer domain, and the ow within the outer domain can be considered irrotational. The second assumption is that the compressibility e ects remain small enough so that the incompressible ow approximation holds for all but the acoustical part of the ow. The usual way to introduce this assumption is to consider that the Mach number, M , is low. If, in addition, density variations are small, this assumption will result in a decomposition of the problem into three sequential problems. The low-Mach-number assumption is used together with the Helmholtz decomposition of the velocity eld in order to obtain two di erent equations for the incompressible and compressible 349
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part of the ow. The latter is assumed to be small enough to justify further linearization based on neglecting terms O(M 2 ) and higher. This results in the two following problems: (a) the incompressible Navier-Stokes equations: @U + (U r)U + 1 rp = U o
o
@t
rU =0 where po and o denote the mean pressure and density of the uid, respectively, and, (b) Lighthill's [8] acoustic pressure equation: 1 @ 2 p = p +
c2 @t2
@ 2 Tij
@xi @xj
(1)
where Tij
= o ui uj + (po ? c2 o )ij
Since the ow is considered isentropic, Lighthill's tensor Tij becomes: Tij
= o ui uj
In the inner domain, the Navier-Stokes equations can be replaced by the vorticity transport equation: @! + (U r)! = (! r)U + ! (2) @t Compressibility e ects are present only in the acoustical part of the ow, i.e. the outer region. According to our assumptions, the ow eld in the outer region is irrotational, and can therefore be described by means of a velocity potential. This suggests a domain decomposition of the ow. From a numerical point of view, the far- eld assumptions allow us to concentrate our computational resources in the inner region where sound is generated. In the outer domain, the computational e ort is greatly reduced by the use of Lighthill's linear inhomogeneous wave equation. However, imposition of adequate boundary conditions must be carefully addressed. Since we have a linear equation, an easy way to handle this problem is to rely on an integral formulation which implicitly satis es the external boundary conditions. The most commonly-used integral representations are based on the linear relation between the acoustic pressure and the time derivative of the velocity potential. The latter can be expressed by means of an integral formulation using the vorticity eld; thus, an integral representation of the pressure eld based on the vorticity distribution is obtained. Moreover, further simpli cation will be obtained by considering that the distance between the support of vorticity and the observation point is large. As a result, the integral representation for the far- eld pressure will be written in a numerically tractable form. We rst solve Lighthill's equation by using the Green's function of the wave equation, o (t
G (x; t) = 4
? ? jx?c yj ) = o (tret ) jx ? yj 4 jx ? yj
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Using this expression leads to the following integral representation for p: Z Z Z Z o @ 2 vi vj (tret ) p(x; t) = dyd (3) 4 D! @yi @yj jx ? yj The next step is to use an asymptotic method based on the previous assumptions that the characteristic length of the inner domain, l, is small compared to the acoustic wavelength, = O(l=M). As a result, jxj will be large compared to jyj in the integral and jx ? yj?1 can be approximated by jxj?1 . Moreover, since jxj >> l, we can use the following approximation: @ @xi
f (t
Consequently, Eq. 3 takes the form:
? jxc j ) ' ?cjxxji @t@ f (t ? jxc j ) 2
x = 4c2ojxj3 @t@ 2
p( ; t)
Z Z Z
D!
(x y)x (! U)dy
(4)
The main interest of the above approximation is that the time integral has been replaced by a time derivative which is an instantaneous quantity, thus easier to compute. Mohring [9] applied the far- eld approximation to derive a simpler form of Eq. 4:
x = ?12o xc2i xr3j dtd 3 Qij (t ? cro ) o 3
p( ; t)
where
Qij =
Z Z Z
D!
y !)j d3 y
yi (
is the second moment of vorticity.
2 Numerical methods 2.1 Vortex methods
The vortical part of the ow is simulated using either a vortex particle method [2] or a vortex lament method [6]. These methods have a strong common basis and di er primarily by the way the stretching term is accounted for. The vortex particle method simulates the following system: @! + (U r)! = (! r)U + ! (5) @t
U(x; t) = 41
Z Z Z
D!
!(y; t) (x ? y) dy
jx ? yj3
(6)
while application of the lament scheme is restricted to inviscid ows. On the other hand, both methods are based on the representation of the ow in terms of vorticity-carrying particles, speci ed by two vector quantities: the location and the amount of transported vorticity,
Xi =
i =
R
RPi
xdv
P dv Z i Pi
!dv ESAIM: Proceedings, Vol. 1, 1996, pp. 349{362
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respectively. The velocity eld is reconstructed from the discrete particle vorticities using a regularized version of the Biot-Savart integral, Eq. 6. A simple and convenient form is given by: X U(x; t) = 41 (jx ?i X(jx6 ?+ X6i))1=2 (7) i i where is a small regularization parameter whose value has to be xed with respect to the averaged distance between two particles. While many di erent forms of the regularized velocity eld have demonstrated their ability to lead to accurate results, it is not yet clear whether one form has obvious advantages over others. For brevity, we sketch the implementation of both schemes to simulation of the inviscid Helmholtz equation. In the particle representation, we rely on an integral formulation of the vortex stretching term, (! r)U, which is obtained by straightforward di erentiation of Eq. 5. This leads to the following set of ordinary di erential equations for the vorticity and location of the particles:
Xi = U(Xi ; t)
d
dt
i = 3
d
dt
X
1
8 j jXi ? Xj j5 [((Xi ? Xj ) i )((Xi ? Xj ) ^ j ) + (((Xi ? Xj ) ^ j ) i )(Xi ? Xj )]
Application of the lament scheme is based on ordering Lagrangian particles along vortex tubes and using the circulation of the vortex tubes to characterize the vorticity of the elements. Thus, the particle vorticity is approximated by the product of the circulation of the vortex tube and the length of the material segment associated to the particle, i.e.
i(t) ' ?li(t)
Using the above approximation, the velocity eld is obtained again from (7). In addition, Kelvin's theorem is invoked to conclude that the circulation of vortex tubes remains unchanged while changes in the material length li (t) are approximated from the deformation of the geometry of the vortex tube. Thus, direct simulation of the vorticity transport equation and evaluation of the stretching term are avoided, at the expense of small additional overhead associated with book-keeping of the geometry of tubes and computations of segment lengths [7].
2.2 Acoustic pressure computation
The main advantage of the domain decomposition method is that there is no feed back e ect of the far- eld ow on the vortex ow. The acoustic pressure can be computed after the incompressible calculations are completed, provided that sucient information is stored. Using the particle discretization of the vorticity eld, Eq. 4, the far- eld acoustic pressure is approximated as:
x = 4c?2joxj3 @t@ 2 2
p( ; t)
X
i
(x Xi )x ( i Ui )dy
With the lament representation, Mohring's formula is used; it yields: 3
x = 12o cx2 jixxjj3 ddtQ3ij
p( ; t)
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where Qij
=
X
m
353
(?m [Xm ]i [lm Xm ]j )
The time derivatives in the above equations are estimated using nite-di erence schemes.
3 Interaction between two vortex rings In this section, we consider the ow generated by two interacting coaxial vortex rings. The initial con guration is illustrated in Fig. 1. When their circulations have identical signs, the motion of the two vortex rings is periodic, with each vortex ring passing through the other during one period. This will be referred to as the mutual threading problem. When the ciculations are of opposite sign, the vortex rings undergo a head-on collision and the ow maintains its symmetry. Both problems have been studied by Kambe [4, 5] who gave theoretical as well as experimental sound predictions, thus providing good test cases for our aeroacoustic model.
3.1 Mutual threading
The mutual threading of two vortex rings may be summarized as follows. The two rings have the same circulation and their induced velocity eld is such that the speed of the rear vortex is larger than that of the front one. As a result, the radius of the trailing vortex ring becomes smaller and threads through the leading vortex. Consequently, the axial position of the two vortices is inversed and a periodic motion is established. Below, results of our simulations are compared with the theoretical predictions of Kambe. In Fig. 2 the ring radius is shown against the distance between the two vortex rings for three di erent initial separations. Each of the three curves represents the position of the front vortex ring during a complete turn. The time needed to complete one turns de nes the period T , which is used to normalize time. The evolution of the separation distance between the two rings is plotted in Fig. 3. The translational speed of the mean axial position of the two rings is non-uniform (Fig. 4). The observed frequency is twice the frequency of the separation of the rings. There are small di erences between our results and those of Kambe. They are probably due to numerical errors as well as to errors in recovering Kambe's curves. The sound emitted by this interaction is associated with the non-uniformity of the axial motion of the rings. The evolution of the third time derivative of Qij is given in Fig. 5. It has the same frequency as the speed of the mean axial position and depends strongly on the initial separation of the rings. The peak values of our results are higher than those of Kambe. This is due to the di erence observed between the predictions of the speed of the mean axial position as shown in Fig. 4. Experimental data are not available but the agreement with the theoretical analysis of Kambe is good, giving con dence in the numerical model.
3.2 Head-on collision
During the collision process, a deceleration in the motion of the two vortex rings occurs as the rings begin to approach each other. Meanwhile, their radii increase rapidly. Computed results are presented for the following conditions, which coincide with the experimental conditions of Kambe [5]: Initial vortex ring radius R = 0:0054m, initial translation velocity U = 57m/s, observer's distance jxj = 0:5 m.
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Figure 6 shows computed pressure pro les together with experimental data obtained by Kambe [4]. Also shown are inviscid calculations of Kambe based on asymptotic approximations and the calculations that he obtained by means of a viscous correction. The favorable agreement between our predictions and Kambe's measurements [4] shows that the numerical model reproduces the basic features of sound emission. The quantitative agreement between computed predictions and experimental data should be emphasized, especially since vortex methods are generally expected to over-predict sound levels. Further examination of Fig. 6 shows that results of the present model lie between Kambe's inviscid and viscous curves. This suggests that there may be need for improving the modelling of di usion e ects in the simulations. However, Kambe mentions that an adjusting parameter must be added so as to get a good t with the observations. This indicates that di usion is coupled to internal motions within the core of the vortex ring and an ecient model should include an accurate description of the core structure evolution. The role of the core structure can be examined by comparing acoustic pressure predictions obtained by xing the initial core structure and varying the initial distance d (Fig. 7). At early stages, the necessary adaptation of the initial conditions results in oscillations which are similar for all the cases considered. Later on, as the rings are approaching each other, this similarity vanishes, indicating that the small di erences observed in the early stages are now growing due to strong interactions between the two vortex rings. Moreover, this can not be corrected by a simple rescaling of time. This indicates that the problem is very sensitive to initial conditions which, at the experimental level, can not be accurately known.
3.3 Non-axisymmetric collisions
Additional insight into the role of core structure variation is gained by considering the impact of: (a) short-wave unstable perturbations and (b) long-wave perturbations with characteristic wavelength much larger than the vortex core. Figure 8 shows the evolution of one of the 3D rings during the collision process. At the start of the computation, the rings are perturbed (from an otherwise axisymmetric state) at the most unstable azimuthal bending mode [11, 6]. For the present conditions, 12 sinewaves of small amplitude are used. The simulation is performed using a lament-based scheme which maintains the 12-fold azimuthal symmetry of the initial con guration in order to speed up the computations. A comparison between axisymmetric and 3D far- eld sound predictions is made in Fig. 9. In both axisymmetric and 3D computations, a xed quadrupolar far- eld directivity is obtained while, as shown in Fig. 9, very small di erences in acoustic pressure levels are detected. The small impact of unstable bending waves on far- eld sound emission may be explained by verifying that the absolutely unstable growth of the waves has very little impact on the time evolution of second moments of vorticity and, consequently, on far- eld sound. The computations above are nally contrasted with results for slender rings perturbed using long waves. Fig. 10 shows the evolution of a thin vortex ring at early stages of a head-on collision evolution event. The two vortex rings are initially perturbed using small-amplitude helical waves with wavenumbers 1 and 2. Note that, in the present case, the perturbation waves travel around the circumference of the ring without signi cant change in their form or amplitudes. Nonetheless, as shown in Fig. 11, the evolution of the waves has strong impact on far- eld sound emission. In particular, it leads to signi cant departures from axisymmetric computations which, at similar stages in the collision, essentially predict zero far- eld acoustic pressure. Also note that, while still quadrupolar in nature, the directivity of the far- eld pressure distribution is no longer timeindependent. These results amplify our earlier observation that accurate predictions of far- eld noise should include adequate representation of the relevant modes of large-scale vortex motion as ESAIM: Proceedings, Vol. 1, 1996, pp. 349{362
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well as internal core structure variation.
4 Example of a complex problem: wind turbine noise 4.1 Wind turbine ow
A wind turbine is made of three distinct parts: a rotor, a nacelle which hosts the electromechanical device, and a tower. From an aerodynamic point of view, in the ow around a wind turbine we can distinguish three regions: (a) the wake region, which extends downstream of the turbine and contains the vorticity which is shed from the blades and the tower, (b) the rotor region, which contains the blades and the tower, and (c) the solid surface region corresponding to the turbulent boundary layers which develop over the blades and the tower. The lowest scales of motion are those of the third region, and a direct description of the ow using the same model for the whole ow eld domain would be impossible. One way to deal with this problem is to adopt di erent methods suitable to describe the ow in the three di erent regions. The model used herein is based on an inviscid ow approximation which restricts the impact of viscous e ects along separation lines on the tower and the blade wherefrom vorticity is released in the free- ow. The boundary layer is assumed to remain attached elsewhere, and the ow due to the presence of solid-walls is described with lifting surface models. The wake region is discretized by means of vortex-carrying particles according to the method described in Section 3. Finally the wind in ow is assumed to be frozen, i.e. the impact of the wind turbine and its wake on the external vortical ow is neglected. The model has been extensively validated for wind turbines [10]; sample results for a wind turbine showing the computational particles in its wake are provided in Fig. 12.
4.2 Sound generation: A \noisy" line model
Sound generated by wind turbines is not limited to low-frequency vortex noise. The main noise source is due to the turbulent ow uctuations developed over the blades. So an accurate description of the relevant phenomena would require computation of the compressible viscous turbulent
ow around the blades. It has been previously mentioned that such a computation is impossible, and that a rough model has to be introduced. In this work, the characteristics of noise sources within boundary layers are related to the vortical ow by means of empirical laws. Based on the 3D inviscid calculations the overall ow characteristics over the blades and tower are determined (e ective angle of attack, reference velocity, etc.). Next these data are used as input in 2D boundary layer calculations. Finally the boundary layer characteristics resulting from these computations are correlated (through appropriate scaling) to the empirical expressions which Brooks et al. [1] obtained by tting with wind tunnel measurements. The results are then incorporated into the model by introducing sources along the blade in a similar fashion to the well-known lifting line model in which the aerodynamic forces are replaced with noise generators. This model was used to predict the noise generated by a wind turbine equipped with rotors having changeable blade tips. The aim of that work was to evaluate di erent blade designs and complement to experimental measurements. Comparison between numerical and experimental results was favorable. A sample of the results is presented in Fig. 13. In all cases, the trends observed in measurements were recovered by the simulations [3]. In particular, experimental and numerical results predicted the same most-silent blade design.
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5 Conclusion
In the present paper some of the questions related to the computation of vortex sound have been considered at di erent levels of complexity. Vortex particle and vortex lament approximations have been used to devise numerical methodologies which enable the study of low-Mach number noise emissions from two interacting vortex rings. The results obtained are considered encouraging since, on one hand, they clearly show the signi cant capabilities of vortex methods in analyzing vortex sound, and, on the other, they highlight areas which need improvement in the models. Of particular importance are the remarks concerning core structure evolution, which is found to play a signi cant role when vortex interactions are intense. At the level of engineering applications, the model was extended to include also high frequency noise sources originating from turbulence. Application to the case of a wind turbine con rmed that in spite of the simpli cations, consistent predictions can be obtained which can guide the correct design of wind turbine blades.
Acknowledgments
The authors wish to acknowledge helpful discussions with Prof. D. Juve. Some computations were performed at the Pittsburgh Supercomputer Center.
References
[1] Brooks, T.F., Pope, D.S. and Marcolini, M.A., Airfoil Self Noise and Prediction, NASA Reference Publication 1218, 1989. [2] Choquin, J.P. and S. Huberson, S., \Interaction Between Numerical and Physical Instabilities in Vortex Flows," Int'l. J. Num. Meth. Fluids, 11, pp. 541{553. [3] Huberson, S., Pothou, K.P. and Voutsinas, S.G., Wind Turbine Noise: Recent Developments of a Vortex Type Aeroacoustic Numerical Tool, EWEC'94, Thessaloniki, 1994. [4] Kambe, T., \Acoustic Emission by Vortex Motions," J. Fluid Mech., 173, pp. 643{666, 1986. [5] Kambe, T. and Minota, T., \Acoustic Wave Radiated by Head-on-Collision of Two Vortex Rings,", Proc. Roy. Soc. London, A386, pp. 277{308, 1983. [6] Knio, O.M. and Ghoniem, A.F., \Numerical Study of a Three-Dimensional Vortex Method," J. Comp. Phys., 86, pp. 75{106, 1990. [7] Leonard, A., \Computing Three-Dimensional Incompressible Flows with Vortex Filaments," Ann. Rev. Fluid Mech., 17, pp. 523{559, 1985. [8] Lighthill, M.J., \On Sound Generated Aerodynamically I. General Theory," Proc. Roy. Soc. London, 222A (1107), pp. 564{587, 1952. [9] Mohring, W., \On Vortex Sound at low Mach Number," J. Fluid Mech., 85, pp. 685{691, 1978. [10] Voutsinas, S.G., Belessis, M. and Huberson, S., Dynamic In ow E ects and Vortex Particle Methods, ECWEC'93, Travemunde, 1993. [11] Widnall, S.E. and Tsai, C.-Y., \The Instability of the Thin Vortex Ring of Constant Vorticity," Proc. Roy. Soc. London, 287 (A1334), pp. 273{305, 1977. ESAIM: Proceedings, Vol. 1, 1996, pp. 349{362
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x
theta 2R
O z
Z
Figure 1: Initial conditions for the problem of two interacting vortex rings.
Figure 2: Ring radius evolution against the distance between the two vortex rings. Z12 is the distance between the two rings and R1 is the rst ring radius. R0 is the radius of both rings at time t = 0.
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Figure 3: Evolution of the separation distance between the two rings. the two rings. R0 is the radius of both rings at time t = 0.
Z12
358
is the distance between
Figure 4: Translational speed Z 0 of the mean axial position of the two rings.
Figure 5: Evolution of the third time derivative of Qij is given versus time.
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Acoustic Pressure 2 1.5
predictions measurements Kambe inviscid Kambe viscous
1
P [Pa]
0.5 0 -0.5 -1 -1.5 -2 1400 1600 1800 2000 2200 2400 2600 2800 3000 Time [micro s]
Figure 6: Computed pressure pro les together with experimental data obtained by Kambe [4].
Figure 7: Acoustic pressure signals obtained using a deformable core model and di erent initial separation distance, d=R = 4 (solid), d=R = 3 (short dash), and d=R = 2 (long dash).
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Figure 8: Evolution of one of the 3D rings during the collision process.
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Figure 9: Comparison between axisymmetric (continuous) and 3D (dotted) far- eld sound predictions.
Figure 10: Evolution of a thin ring at early stages of a head-on collision. The plots show projections of the ring centerline on an azimuthal plane (left) and a streamwise plane (right) at di erent times.
Figure 11: Evolution of the far- eld acoustic pressure during the early stages of collision between perturbed slender vortex rings. The solid line depicts emission along the axis of propagation of the ring, while the dashed lines show emission along two orthogonal directions in a streamwise plane.
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Vw
Figure 12: Computational particles in the wake of a wind turbine.
cref = 0.2025m
R=8.13m R=7.11m
UNIWEX WT: Comparison between Ogee and Reference blade tips 60 Reference-Predictions Ogee-Predictions 55 Reference-Measurements Ogee-Measurements 50
R=8.00m OGEE tip
cref
reference tip
SPL_1/3 [dB]
shark fin tip
45 40 35 30
cref
original blade
1.5 cref
25 20
5.50
6.00
6.50
7.00
7.50
8.00
8.50
9.00
15
R [m] 10 800 1000
2000 4000 Frequency [Hz]
6000 8000
Figure 13: (left) Di erent blade tip shape. Only the Ogee and reference tip have been used in the present calculations. (right) Experimental and numerical pressure for di erent blade designs.
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